Rationalizing two-tiered choice functions through conditional choice

Set-valued choice functions provide a framework that is general enough to encompass a wide variety of theories that are significant to the study of rationality but, at the same time, offer enough structure to articulate consistency conditions that can be used to characterize some of the theories within this encompassed variety. Nonetheless, two-tiered choice functions, such as those advocated by Isaac Levi, are not easily characterized within the framework of set-valued choice functions. The present work proposes conditional choice functions as the proper carriers of synchronic rationality. The resulting framework generalizes the familiar one mentioned above without emptying it and, moreover, provides a natural setting for two-tiered choice rules.     

Choice probabilities and choice functions

Choice probabilities are basic to much of the theory of individual choice behavior in mathematical psychology. On the other hand, consumer economics has relied primarily on preference relations and choice functions for its theories of individual choice. Although there are sizable literatures on the connections between choice probabilities and preference relations, and between preference relations and choice functions, little has been done—apart from their common ties to preference relations—to connect choice probabilities and choice functions. The latter connection is studied in this paper. A family of choice functions that depends on a threshold parameter is defined from a choice probability function. It is then shown what must be true of the choice probability function so that the choice functions satisfy three traditional rationality conditions. Conversely, it is shown what must be true of the choice functions so that the choice probability function satisfies a version of Luce's axiom for individual choice probabilities.     

Coherent choice functions under uncertainty

We discuss several features of coherent choice functions—where the admissible options in a decision problem are exactly those that maximize expected utility for some probability/utility pair in fixed set S of probability/utility pairs. In this paper we consider, primarily, normal form decision problems under uncertainty—where only the probability component of S is indeterminate and utility for two privileged outcomes is determinate. Coherent choice distinguishes between each pair of sets of probabilities regardless the “shape” or “connectedness” of the sets of probabilities. We axiomatize the theory of choice functions and show these axioms are necessary for coherence. The axioms are sufficient for coherence using a set of probability/almost-state-independent utility pairs. We give sufficient conditions when a choice function satisfying our axioms is represented by a set of probability/state-independent utility pairs with a common utility.     

Deviance and a generalized disposition toward internality: An attributional approach

Two studies were carried out to test the hypothesis, derived from attribution theory, that adolescents might acquire an identity and mediating beliefs about reality (internal locus of control) by deviating from norms when they perceived adults to be powerful. In line with this expectation, the major study found that students rated by teachers as "deviant," but not those rated as "conformant," showed greater internality in beliefs about man (control ideology), p = .024, under conditions of high power. Despite the following limitations or exceptions, attribution theory seemed to best explain the predicted effect: (1) the effect for beliefs about control was nonsignificantly weaker for females then males, perhaps because of different meanings of deviance for males and females and female socialization for conformity rather than independence; (2) in the major study the effect was absent for questions dealing with respondents' personal feelings of control ("self-attributions"); and (3) in both studies there was a tendency for deviant females to show more personal externality (relative to conformers) when perceived power was high.     

Choice between fixed-interval schedules: Graded versus step-like choice functions

Pigeons chose between two fixed-interval schedules of food reinforcement. A single peck on one of two lighted keys started the fixed-interval schedule correlated with that key. The schedule had to be completed before the next choice opportunity. The durations of the fixed intervals were varied over conditions from 15 s to 40 s. To maximize the rate of reinforcement, the pigeons had to choose exclusively the shorter of the two schedules. Nevertheless, choice was not all-or-none. Instead, relative choice, and the rates of producing the fixed intervals, varied in a graded fashion with the disparity between the two schedules. Choice ratios under this procedure (single response to choose) were highly sensitive to the ratios of the fixed-interval schedules.     

Discriminative functions of schedule stimuli and memory: a combination of schedule and choice procedures

Pigeons responded under a combination brief-stimulus schedule and choice procedure. Normally, a fixed-interval schedule was in effect, where completion randomly produced either a brief stimulus or food. Intermittently, this schedule was interrupted by a choice arrangement. Two choice keys were lit, either a short or a long time since a prior event (food or stimulus). One choice response produced food if the time had been short, and the alternate response produced food if the time had been long. Across conditions, the duration of the fixed-interval schedule was varied, the stimuli that comprised the brief-stimulus operation were changed, and the stimuli were presented as paired and nonpaired with food. The focus of the study was the control of both schedule performance and choice responding across conditions. The results showed that choice accuracy was correlated with the degree of fixed-interval curvature, the response pattern of a pause followed by a gradually accelerated rate. As fixed-interval schedule duration was increased, both the degree of fixed-interval curvature and choice accuracy decreased. The particular brief stimulus used affected schedule and choice performance, with a more salient stimulus producing a greater degree of curvature and higher accuracy. Pairing and nonpairing operations produced striking differences in performance with the less salient brief stimulus, but not with the more salient stimulus. The results suggest that brief-stimulus schedule performance may be conceptualized in the context of memory research.     

An adaptive approach for modelling multiattribute choice

An adaptive approach for modelling individual-level choice among multiattribute alternatives using the binary logit model is presented. The algorithm involves the collection of paired comparison data. In an effort to maximize the amount of information obtainable from each response, it is based on the experimental design criterion of D-optimality. A simulation study indicates that the proposed algorithm outperforms other sequential selection approaches in terms of estimation accuracy and predictive efficiency under certain circumstances. The results appear to encourage the use of such an adaptive algorithm for individual-level modelling in light of the potential reduction in data requirements without significant loss in predictive accuracy.     

Speed-accuracy tradeoff functions in choice reaction time: Experimental designs and computational procedures

The existence of tradeoffs between speed and accuracy is an important interpretative problem in choice reaction time (RT) experiments. A recently suggested solution to this problem is the use of complete speed-accuracy tradeoff functions as the primary dependent variable in choice RT ,experiments instead of a single mean RT and error rate. This paper reviews and compares existing procedures for generating empirical speed-accuracy tradeoff, functions for use as dependent variables in choice RT experiments. Two major types of tradeoff function are identified, and their experimental designs and computational procedures are discussed and evaluated. Systematic disparities are demonstrated between the two tradeoff functions in both empirical and computer-simulated data. Although all existing procedures for generating speed-accuracy tradeoff functions involve empirically untested assumptions, one procedure requires less stringent assumptions and is less sensitive to sources of experimental and statistical error. This procedure involves plotting accuracy against RT over a set of experimental conditions in which subjects’ criteria for speed vs. accuracy are systematically varied.     

The generalized matching law in elite sport competition: football play calling as operant choice

A mathematical model of operant choice, the generalized matching law was used to analyze play-calling data from the 2004 National Football League season. In all analyses, the relative ratio of passing to rushing plays was examined as a function of the relative ratio of reinforcement, defined as yards gained, from passing versus rushing. Different analyses focused on season-aggregate data for the league as a whole, game-by-game data for the league as a whole, and game-by-game data for individual teams. In all analyses except those for a few individual teams, the generalized matching law accounted for a majority of variance in play calling. The typical play-calling pattern reflected undermatching (suggesting imperfect sensitivity of play calling to yardage-gained reinforcers) and a bias for calling rushing plays. Bias was found to be a function of both the relative risk of turnovers and the relative variability in yards gained associated with passing versus rushing plays. The external validity of the matching analyses was supported by significant correlations between parameters of the generalized matching law and team success on offense and season winning percentage. These results illustrate the broad applicability of the generalized matching law to problems outside of the laboratory.     

Magnitude comparison revisited: An alternative approach to binary choice under uncertainty

Two generations of psychologists have been interested in understanding binary choice under uncertainty. In the 1970s and 1980s, researchers assumed that people rely on a two-stage magnitude comparison process to make these decisions (Banks, 1977; Moyer & Dumais, 1978). More recently, the focus has shifted to approaches that rely on probabilistic cues and simple heuristics (Gigerenzer & Goldstein, Psychological Review 103, 650-669, 1996). Here, we test competing predictions derived from these two very different approaches and conclude that the magnitude comparison process plays a central role in this task. In support of this conclusion, we present an experiment in which participants were timed as they decided which of two vehicles was more expensive. Pairs composed of one luxury vehicle (e.g., BMW 323i) and one nonluxury vehicle (e.g., Toyota 4Runner) were critical because the magnitude comparison approach correctly predicted that reaction times would decrease with subjective distance, whereas the heuristics approach incorrectly predicted that there would be no relation.     

Preference-based escalation: A new interpretation for the responsibility effect in escalating commitment and entrapment

A prominent finding in escalating commitment and entrapment research is the “responsibility effect”: people invest more in a losing course of action or persist with it for longer if they themselves initiated this action (responsibility) as opposed to if it was assigned to them. We argue that this effect is driven by participants’ preferences. Responsible participants usually prefer the chosen alternative since they have chosen it themselves. Non-responsible participants, in contrast, represent a mix of persons who either favor or disfavor the chosen alternative. In two experiments, we demonstrate that responsible participants favor the chosen course of action more strongly than non-responsible participants do, that these preferences facilitate reinvestment in and persistence with the chosen course of action, and that responsibility has no effect over and above this effect of preferences. Non-responsible participants preferring the chosen course of action made similar reinvestments and exhibited similar persistence as responsible participants.     

Decision biases in intertemporal choice and choice under uncertainty: Testing a common account

Two experiments were performed to test a psychophysical account of parallels between biases in risky choice and intertemporal choice. Experiment 1 demonstrated the common difference effect in intertemporal choice and the common ratio effect in risky choice. As was predicted, these two biases were uncorrelated with each other, although each was correlated across monetary/health domains. This result is consistent with the supposition that these two biases result from psychophysical properties of two different dimensions (time and probability, respectively). Experiment 2 examined the magnitude effect in intertemporal choice and the peanuts effect in risky choice. These two biases were correlated with each other but were uncorrelated across monetary/health domains. This result is consistent with the supposition that these two biases result from psychophysical properties of the same dimension (utility of money or health).     

Predicting from expressed vocational choice: a review

Information on the predictive validity of expressed vocational choice was gathered in a review of large-sample longitudinal studies. These results indicate that, in general, a person's expressed vocational choice predicts his future employment about as well as interest inventories or combinations of personality and background characteristics. Expressed vocational and educational field choices are moderately stable over one- to five-year periods during and after college. Difficulties and shortcomings in past research are discussed, and recommendations are made for future research on the predictive validity of expressed choice. Implications of these results for the practicing counselor are also considered.     

Transforming illness into choice: a spiritual perspective

Illness is seen as the enemy in today's world, while spirituality might be considered counter-cultural. This article explores the positive attributes of illness seen through the eyes of spirituality. By identifying spiritual themes in a person's life, especially evident in times of illness, one may maximize the experience of illness for its benefit rather than its curse.     

Spaced responding and choice: a preliminary analysis

Pigeons were exposed to reinforcement both for short (2 < IRT < 3 sec) and long (10 < IRT < 11 sec) interresponse times. They developed bimodal interresponse-time distributions, which were decomposable into two independent component distributions under the control of the short and long contingencies respectively. The birds' allocation of responses between these two distributions was determined by a simple power-law relationship between reinforcement ratios, and response ratios derived from the component distributions. Comparison between this situation and concurrent choice situations raises the possibility that the power-law relation between ratios may be a more general law of choice than the matching of relative frequencies (probabilities).